For every given set of parameters, we solve for the equilibrium defined in Definition 5.1 of the main text by iterating on the equilibrium conditions pre- and post-reform. This allows us to simulate aggregate variables as well as policy functions contingent on being in a firm about to close down. Using these policy functions, we then simulate H data sets of size
N0 and N1 for the period before and after the reform, respectively, and estimate the RDD
regressions and macro moments based on them.
Denote by β ≡ (λ, σε, ξ, m0, c) the parameters to be estimated and ˜m(β, e) the simu-
lated moments given parameters β and a set of shocks e. Call m the corresponding vector of empirical moments. We choose β by solving
min
β ( ˜m(β, e) − m)
0
W ( ˜m(β, e) − m) ,
where W is a weighting matrix. Efficient GMM requires setting W equal to the inverse of
variance-covariance matrix of m. Instead, we choose W equal to the identity matrix.1 The
reason is that we view the model rather as a description of firms about to experience a mass layoff than as a representation of the entire economy. We include macro moments to ensure that the parameters are identified and not unrealistic but do not expect the model to reproduce macro moments and to be perfectly in line with aggregate data solely using data from firms experiencing negative shocks. In other words, we do not expect the parameters describing the behavior of these firms to be perfectly in line with the entire economy. Since the macro moments are measured with much less variance, efficient GMM will put much more weight on the macro moments, while our primary interest lies in explaining worker mobility in declining firms.
A well known problem with Method of Simulated Moments is that the simulated mo- ments are a discontinuous function of the underlying parameters for a given set of random shocks, as we have a finite number of observations and discrete outcomes. This can pose problems to optimization algorithms, leading to non-convergence or convergence to local optima. Keane and Smith (2003) propose a remedy for this problem in the context of a
random utility model: Suppose that a binary variable yi is 1 if a simulated latent utility
given parameters β, ui(β), is positive and zero otherwise. Instead of using yi to calculate
the simulated moments, they propose using a continuous function of the latent utility,
g(ui; ζ), where g(ui; ζ) → yi as ζ → 0. Our choice for g(u; ζ) is
g(u; ζ) = Φ(u/ζ),
where Φ(·) denotes the c.d.f. of the standard normal distribution. Paralleling this strategy, we apply this smoothing procedure to all discrete outcomes of the model, i.e. to the policy
1We only give weight 0.1 to expected hiring costs, as this is the only moment not based on Austrian
functions γ0(p), γ1(p), µ0(p, po), and µ1(p, po).
There is no clear rule as to which value should be chosen for the smoothing parameter ζ and the number of simulated data sets H. Larger values of ζ and H lead to a smoother surface of the objective function, decreasing the risk of local optima where the optimization algorithm could get stuck. At the same time, increasing ζ increases the bias, while a higher H is more computationally expensive. For the results reported here, we choose H = 5, which still leads to manageable computation time. We then chose ζ so that the objective
function is reasonably smooth.2
In Figure 10, we plot the stationary productivity distributions before and after the reform and the change in the number of matches in every productivity bin. Clearly, we observe a rightward shift as less productive firms exit the market more quickly, while more productive firms benefit from the higher arrival rate of workers. In reality, this effect might
even be stronger as we have reason to believe that our model underestimates this effect.3
Figure 10: Stationary productivity distribution before and after the reform
Number of Matches 0 0.02 0.04 0.06 0.08 Pre-reform Post-reform Log Productivity -0.1 -0.05 0 0.05 0.1
Change due to Reform
#10-3 -4 -2 0 2 4
Note: The top plot shows the stationary number of workers of either eligibility status (i.e. n0(p) + n1(p))
per productivity state implied by the model using calibrated (Table 4 in the main text) and estimated (Table 5 in the main text) model parameters, distinguishing between the pre- and post-reform steady state. The bottom plot shows the difference in the number of workers per productivity state between the post- and the pre-reform steady state.
2
We choose ζ as low as possible. It turned out that ζ = 0.2 resulted in a reasonably smooth surface.
3To keep the model manageable for our structural estimation, we assume that entrepreneurs that exit
the market get to draw from the stationary distribution again. Thus, a very bad entrepreneur might re-enter as a very good one.
Figure 11: Cumulative share of workers leaving into unemployment and to another job, simulated values vs. data (allowing for temporary employment)
(a) Cumulative exit rates (all exits)
Distance to shock (months)
-20 -15 -10 -5
Proportion having left firm 0 0.1 0.2 0.3
0.4 All exits
Data: Old system Data: New system Model: Old system Model: New system
(b) Cumulative exit rates (JTJ)
Distance to shock (months)
-20 -15 -10 -5
Proportion having left firm 0 0.05 0.1 0.15
0.2 Job-to-Job
Data: Old system Data: New system Model: Old system Model: New system
(c) Cumulative exit rates (JTU)
Distance to shock (months)
-20 -15 -10 -5
Proportion having left firm 0 0.05 0.1 0.15
0.2 Job-to-Unemployment
Data: Old system Data: New system Model: Old system Model: New system
Note: The Figures compare cumulative exit rates implied by an RDD (see main text for details) on our baseline sample to an RDD based on artificial data generated by the model allowing for temporary jobs using calibrated (Table 4 in the main text) and estimated (Table 5 in the main text) model parameters. Figure (a) shows all exits, while figures (b) and (c) distinguish between JTJ and JTU transitions. Note that we only fit figures (b) and (c) as figure (a) corresponds to the sum of figures (b) and (c).